Where We’ve Been…….
Properties, Part I……..

Where We Are Going…….
Just listen.
Today, we are going to investigate one of the most important properties you will use this year and in future classes.
Algebra
Associative Properties Commutative Properties
Distributive Property
Identity Properties Order of Operations Translating Expressions

The Distributive Property
6.6 p. 485

What Is Our Objective? p.485
• Use the distributive property to rewrite and simplify equivalent multiplication problems
Three friends went to a baseball game. Each ticket cost $20 and all three friends bought a baseball hat for $15 each.
Notes.
Let’s look at this expression: 3(20 + 15).
The 3 represents:
The 20 represents:
Three friends
Price of the ticket
The 15 represents:
Price of the hat

Three friends went to a baseball game. Each ticket cost $20 and all three friends bought a baseball hat for $15 each.
2. Let’s evaluate the expression…write the 3 in front of the parentheses……
3(20 + 15) =
3 x 35 = 105
3. What does the expression 3 x 20 + 3 x 15 represent?
3 x 20 represents
3 x 15 represents
The price of three tickets
The price of three hats
4. Evaluating this expression:
3 x 20 + 3 x 15
3 x 20 = 60
3 x 15 = 45
60 + 45 = 105
What do you notice about the answers of these two problems?

WAIT…..Let’s practice this with just whole numbers first!
Insert this practice page with your other notes for this lesson.
Let’s talk through the first one.
5 x 27
Let’s take the larger factor and write it in expanded form.
5 (20 + 7) Remember, no sign means to multiply!
Let’s “ distribute “ the 5. It is the factor used on both the 20 and the 7.
(5 x 20) + (5 x 7) =
(100) + (35) =
135

4 x 26
Take the larger factor and write it in expanded form.
4 (20 + 6) Let’s “ distribute “ the 4 . It is the value used on both the 20 and the 8.
(4 x 20) + (4 x 6) =
(80) + (24) = 104
7 x 32 = 7 (30 + 2 )
(7 x 30) + (7 x 2) = 210 + 14 =
224

6(50) + 6(6)
300 + 36 = 336
8(70) + 8(3)=
560 + 24 = 584

Back to the text . Example 1.
9 x 4
1
3
=
9 ( 4 +
1
3
) = 9(4) + 9(
1
3
)
36 + 3 = 39 p. 486

9 ( + )
Use your imagination!
36 + 3= 39
9(4) + 9(1/3)

SHOW YOUR STEPS as you evaluate each of these.
5 x 2
3
5
12 x 2
1
4
=
5 ( 2 +
3
5
) =
Think of 15 ÷ 5
5(2) + 5(
3
10 +
5
)
3 = 13
= p. 486
12 ( 2 +
1
4
) = 12(2)+ 12(
24 + 3 = 27
2 x 3.6 =
1
4
)
2 ( 3 + .6) = 2(3) + 2(
.6)
6 + 1.2 = 7.2

Let’s stop here for today.
• What was the goal of our lesson?
• Tonight you will practice only the process of evaluating with the distributive property.

x
Adding Variables to the Process…
1
1
1 x
1
1
1
2(x + 3) = 2x + 2(3)
= 2x + 6
We will use the distributive property to rewrite the expression that describes this model: 2(x + 3)
Can we do anything else?????
NO!! We do not know the value of x, so we have to stop here.
8(x + 3) 5(9 + x) 2(x + 3)
8x + 8(3)
5(9) + 5x 2x + 2(3)
8x + 24 45 + 5x
2x + 6

Distributive Property II
In our previous lesson, we rewrote expressions in a simpler way using the distributive property.
7 x 35 was rewritten as………
7(30 + 5)
7(30) + 7(5)
210 + 35 = 245
We also used variables.
3 ( b + 5)
3b + 15

3 ( a

4 )

3 a

3 ( 4 )
3 a

12
5 ( b

6 )
 5 b

5 b

5 ( 6 )
30

p. 487 Think Critically
Fran is making a pair of earrings and a bracelet for four friends. Each pair of earrings uses 4.5 cm of wire and each bracelet uses 13 cm of wire. Write two equivalent expressions and then find out how much wire is used.
4(4.5 + 13) and 4(4.5) + 4(13) are equivalent.
4(17.5) = 70 cm 18 + 52 = 70 cm
Sort all of the information you have:
What do you need for one set of stuff????
Earrings: 4.5 cm bracelet: 13 cm
How many are being made? 4

p. 487 Think Critically
Each day, Martin lifts weights for 10 minutes and runs on the treadmill for 25 minutes. Write two equivalent and find the total minutes that Martin exercises in 7 days.
7(10 + 25) and 7(10) + 7 ( 25)
7(35) 70 + 175
245 minutes 245 minutes
7(10) + 7(20) + 7(5) ???????
70 + 140 + 35 105 + 140 = 245

Factoring Expressions
Let’s factor the numerical expression 12 + 8
12 = 2 ·2·3
8 = 2 ·2·2
Ask yourself, “What the largest number 12 and 8 share?”
We know it is 4. OK…..if we divide each of the numbers by 4 – factor the 4 out of each one – what’s left??? Think of the values as multiplication problems with 4’s.
4 is the shared factor…..What do we have left if we just write it once???

14
Let’s look at that again……this is not in your notes.

21

7

2

7

3
7

3 )
Okay. Let’s do the practice problems at the bottom of p. 487.
7 x 2 + 7 x 4 ?????
9 + 21 14 + 28 80 + 56
3 x 3 + 3 x 7
14 x 1 + 14 x 2 8 x 10 + 8 x 7
3
3 ( 3 + 7)
14
14 ( 1 + 2)
8
8( 10 + 7)

Let’s add some variables!
3
3 x x


15
3

5
( x

5 )
16 + 4x 7x + 42 36x + 30
4 · 4 + 4x 7x + 7 · 6
6 · 6 x + 6 · 5
4(4 + x) 7(x + 6)
6(6x + 5)

Let’s try one more of those……….
( n

6 )
3 ( 4 n

6 )
 3 ( 4 ) n

3 ( 6 )
12 n

18

2
1) 9 x 8
2
3
9(8 +
3
)
9(8) + 9(
2
3
)
Guided Practice p. 488
2) 3(x + 1) 3) 5(x + 8)
3x + 3(1) 5x + 5(8)
3x + 3 5x + 40
4) 4(x + 6) 5) 25 + 60 6) 4x + 40
4x + 4(6) 5(5 + 12) 4( x + 10)
4x + 24
Six friends: Admission is $9.50. One ride is $1.50
6(9.5 + 1.5) = 6(9.5) + 6(1.5) = 6(11) = $66

Independent Practice p. 489 Talk to each other. Talk to me.
1. 9 x 44 2. 4 x 5
1
8
9(40 + 4)
9(40) + 9(4)
360 + 36
396
4(5 +
1
)
20 +
4
8
8
4(5) + 4(
1
8
)
= 20
1
2
3. 7 x 3.8
7(3 + .8)
7(3) + 7(.8)
21 + 5.6)
26.6
4. 8(x + 7) 5. 6(11 + x) 6. 8(x + 1)
8x + 8(7)
8x + 56
6(11) + 6x
66 + 6x
8x + 8(1)
8x + 8

Factor Each Expression
8. 8 + 16 9. 54 + 24 10. 63 + 81
8(1) + 8(2) 6(9) + 6(4) 9(7) + 9(9)
8(1 + 2) 6(9 + 4) 9(7 + 9)
11. 11x + 55 12. 32 + 16x 13. 77x + 21
11x + 11(5) 16(2) + 16x
7(11)x + 7(3)
11(x + 5)
16(2 + x) 7(11x + 3)

• We have used the distributive property to solve multiplication problems.
• We have written problems with variables using the distributive property.